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## 3. The Neumann Laplacian

1. ### Optimizers for Neumann eigenvalues among convex sets of a fixed perimeter

#### Problem 3.1.

[A. Henrot] Prove the existence of optimizers for $$\sup\{\mu_k(\Omega)\colon \Omega\subset \mathbb{R}^n \mbox{ convex, }P(\Omega)=P_0\}$$ and $$\inf\{\mu_k(\Omega)\colon \Omega\subset \mathbb{R}^n \mbox{ convex, }P(\Omega)=P_0\}.$$
For $k=1$, $n=2$ and convex $\Omega$, R. Laugesen and B. Siudeja have conjectured that $P(\Omega)^2\mu_1(\Omega)\leq 16\pi^2$, where equality is attained if $\Omega$ is either a square or an equilateral triangle (open problem 6.66 in [MR3681143]).
• ### Maximizing the ratio of Neumann eigenvalues among convex domains

#### Conjecture 3.2.

[A. Henrot] For any convex $\Omega\subset \mathbb{R}^n$, it holds that
1. $\mu_2(\Omega)/\mu_1(\Omega)\leq 4$.
2. $\mu_k(\Omega)/\mu_1(\Omega)\leq k^2$.
The two-dimensional version of the above conjecture is formulated by M. Ashbaugh and R. Benguria in [MR1215424]. Partial analytic result again in two dimensions is obtained by P. Antunes and A. Henrot in [MR2795792].
• ### Hot spots conjecture

This conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.

#### Conjecture 3.3.

The first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\Omega$ if
1. $\Omega\subset \mathbb{R}^2$ is simply connected.
2. $\Omega\subset \mathbb{R}^n$ ($n \ge 3$) is convex.
Positive results concern various special classes of domains such as triangles [arxiv:1802.01800], $\mathsf{Lip}$-domains [MR2051611], and thin curved strips [MR3912674]. A counterexample in the class of non-simply-connected planar domains is constructed in [MR1680567].
• ### Directional hot spots conjecture

Let $\Omega \subset \mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.

#### Problem 3.4.

[T. Beck] Can one always find a direction $\mathbf{e}\in\mathbb{R}^2$ such that $$\mathbf{e}\cdot \nabla u >0 \quad \mbox{in }\Omega?$$
An affirmative answer to the above question implies the hot spots conjecture for all centrally symmetric bounded convex planar domains. Under the assumption that $\Omega$ is symmetric with respect to both coordinate axes, the respective question has been affirmatively answered by D. Jerison and N. Nadirashvili in [MR1775736].

Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.